#8635658

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,13,16)(11,14,18,17)(12,15,21,20)(19,22,24,23)(26,30)(31,37)(32,38)(39,40);(1,2,4,5)(3,6,7,8)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,21)(22,24)(26,30)(27,33)(32,38)(34,35);(2,5)(6,8)(11,18)(14,17)(19,24)(22,23)(25,26,29,30)(27,31,33,37)(28,32,36,38)(34,39,35,40);(3,7)(6,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,23)(22,24)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8542854);